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    Ordinal identity guarantees structural isomorphism but no... — Carmelics
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    Challenges→Two infinite well-ordered sets with the same ordinal number have the same cardinal number.

    Ordinal identity guarantees structural isomorphism but not cardinality equivalence when cardinal and ordinal number theories are defined independently.

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    1 reason for
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    Reasons For

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    Reason for
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    • 1.Ordinal structure (successor relations, ordering) can be preserved across systems while cardinality (set size) varies if definitions operate independently.
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    • 2.The natural numbers exemplify this: ordinal sequence (1,2,3...) remains identical under different set-theoretic constructions with different cardinality criteria.
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    • 3.Independent axiomatic systems can define identity conditions separately, making structural isomorphism sufficient for ordinal equivalence without requiring cardinality matching.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Ordinal identity intrinsically depends on cardinality; two sequences differ ordinally if they assign different sizes to their elements or domains.
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    • 2.The claim conflates notational or formal independence with metaphysical independence; definitions being separate doesn't prevent ordinal facts from constraining cardinality facts.
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    Proof of definition segments1 linkedModality & Possibility1 linked

    Related

    Independent axiomatic systems can define identity conditions separately, making ...Ordinal identity intrinsically depends on cardinality; two sequences differ ordi...Ordinal structure (successor relations, ordering) can be preserved across system...The claim conflates notational or formal independence with metaphysical independ...
    +2 moreShow less
    The natural numbers exemplify this: ordinal sequence (1,2,3...) remains identica...Two infinite well-ordered sets with the same ordinal number have the same cardin...

    Details

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    claim
    Perspectives
    2 (1 for, 1 against)
    Edits
    1 edit